Sampling discretization of integral norms. Lecture 3 Vladimir - - PowerPoint PPT Presentation

Sampling discretization of integral norms. Lecture 3

Vladimir Temlyakov Chemnitz; September, 2019

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Sampling discretization with absolute error

Let W ⊂ Lq(Ω, µ), 1 ≤ q < ∞, be a class of continuous on Ω

• functions. We are interested in estimating the following optimal

errors of discretization of the Lq norm of functions from W

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Sampling discretization with absolute error

Let W ⊂ Lq(Ω, µ), 1 ≤ q < ∞, be a class of continuous on Ω

• functions. We are interested in estimating the following optimal

errors of discretization of the Lq norm of functions from W erm(W , Lq) := inf

ξ1,...,ξm sup f ∈W

• f q

q − 1

m

m

• j=1

|f (ξj)|q

• ,

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Sampling discretization with absolute error

Let W ⊂ Lq(Ω, µ), 1 ≤ q < ∞, be a class of continuous on Ω

• functions. We are interested in estimating the following optimal

errors of discretization of the Lq norm of functions from W erm(W , Lq) := inf

ξ1,...,ξm sup f ∈W

• f q

q − 1

m

m

• j=1

|f (ξj)|q

• ,

ero

m(W , Lq) :=

inf

ξ1,...,ξm;λ1,...,λm

sup

f ∈W

• f q

q − m

• j=1

λj|f (ξj)|q

• .

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

General theorem

Theorem (T1; VT, 2018) Assume that a class of real functions W is such that for all f ∈ W we have f ∞ ≤ M with some constant M. Also assume that the entropy numbers of W in the uniform norm L∞ satisfy the condition εn(W , L∞) ≤ Cn−r, r ∈ (0, 1/2). Then erm(W ) := erm(W , L2) ≤ Km−r.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Comments

Theorem T1 is a rather general theorem, which connects the behavior of absolute errors of discretization with the rate of decay

• f the entropy numbers. This theorem is derived from known

results in supervised learning theory. It is well understood in learning theory that the entropy numbers of the class of priors (regression functions) is the right characteristic in studying the regression problem.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Comments

Theorem T1 is a rather general theorem, which connects the behavior of absolute errors of discretization with the rate of decay

• f the entropy numbers. This theorem is derived from known

results in supervised learning theory. It is well understood in learning theory that the entropy numbers of the class of priors (regression functions) is the right characteristic in studying the regression problem. We impose a restriction r < 1/2 in Theorem T1 because the probabilistic technique from the supervised learning theory has a natural limitation to r ≤ 1/2.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Comments

Theorem T1 is a rather general theorem, which connects the behavior of absolute errors of discretization with the rate of decay

• f the entropy numbers. This theorem is derived from known

results in supervised learning theory. It is well understood in learning theory that the entropy numbers of the class of priors (regression functions) is the right characteristic in studying the regression problem. We impose a restriction r < 1/2 in Theorem T1 because the probabilistic technique from the supervised learning theory has a natural limitation to r ≤ 1/2. It would be interesting to understand if Theorem T1 holds for r ≥ 1/2.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Comments

Theorem T1 is a rather general theorem, which connects the behavior of absolute errors of discretization with the rate of decay

• f the entropy numbers. This theorem is derived from known

results in supervised learning theory. It is well understood in learning theory that the entropy numbers of the class of priors (regression functions) is the right characteristic in studying the regression problem. We impose a restriction r < 1/2 in Theorem T1 because the probabilistic technique from the supervised learning theory has a natural limitation to r ≤ 1/2. It would be interesting to understand if Theorem T1 holds for r ≥ 1/2. Also, it would be interesting to obtain an analog of Theorem T1 for discretization in the Lq, 1 ≤ q < ∞, norm.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Smoothness classes

For classes of smooth functions we obtained error bounds, which do not have a restriction on smoothness r. We proved the following bounds for the class Wr

2 of functions on d variables with

bounded in L2 mixed derivative.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Smoothness classes

For classes of smooth functions we obtained error bounds, which do not have a restriction on smoothness r. We proved the following bounds for the class Wr

2 of functions on d variables with

bounded in L2 mixed derivative. Theorem (T2; VT, 2018) Let r > 1/2 and µ be the Lebesgue measure on [0, 2π]d. Then ero

m(Wr 2, L2) ≍ m−r(log m)(d−1)/2.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Marcinkiewicz problem

Let Ω be a compact subset of Rd with the probability measure µ. We say that a linear subspace XN of the Lq(Ω), 1 ≤ q < ∞, admits the Marcinkiewicz-type discretization theorem with parameters m and q if there exist a set {ξν ∈ Ω, ν = 1, . . . , m} and two positive constants Cj(d, q), j = 1, 2, such that for any f ∈ XN we have C1(d, q)f q

q ≤ 1

m

m

• ν=1

|f (ξν)|q ≤ C2(d, q)f q

q.

(1)

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Marcinkiewicz problem

Let Ω be a compact subset of Rd with the probability measure µ. We say that a linear subspace XN of the Lq(Ω), 1 ≤ q < ∞, admits the Marcinkiewicz-type discretization theorem with parameters m and q if there exist a set {ξν ∈ Ω, ν = 1, . . . , m} and two positive constants Cj(d, q), j = 1, 2, such that for any f ∈ XN we have C1(d, q)f q

q ≤ 1

m

m

• ν=1

|f (ξν)|q ≤ C2(d, q)f q

q.

(1) In the case q = ∞ we define L∞ as the space of continuous on Ω functions and ask for C1(d)f ∞ ≤ max

1≤ν≤m |f (ξν)| ≤ f ∞.

(2)

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Marcinkiewicz problem

Let Ω be a compact subset of Rd with the probability measure µ. We say that a linear subspace XN of the Lq(Ω), 1 ≤ q < ∞, admits the Marcinkiewicz-type discretization theorem with parameters m and q if there exist a set {ξν ∈ Ω, ν = 1, . . . , m} and two positive constants Cj(d, q), j = 1, 2, such that for any f ∈ XN we have C1(d, q)f q

q ≤ 1

m

m

• ν=1

|f (ξν)|q ≤ C2(d, q)f q

q.

(1) In the case q = ∞ we define L∞ as the space of continuous on Ω functions and ask for C1(d)f ∞ ≤ max

1≤ν≤m |f (ξν)| ≤ f ∞.

(2) We will also use a brief way to express the above property: the M(m, q) theorem holds for a subspace XN or XN ∈ M(m, q).

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Marcinkiewicz problem with weights

We say that a linear subspace XN of the Lq(Ω), 1 ≤ q < ∞, admits the weighted Marcinkiewicz-type discretization theorem with parameters m and q if there exist a set of knots {ξν ∈ Ω}, a set of weights {λν}, ν = 1, . . . , m, and two positive constants Cj(d, q), j = 1, 2, such that for any f ∈ XN we have C1(d, q)f q

q ≤ m

• ν=1

λν|f (ξν)|q ≤ C2(d, q)f q

q.

(3)

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Marcinkiewicz problem with weights

We say that a linear subspace XN of the Lq(Ω), 1 ≤ q < ∞, admits the weighted Marcinkiewicz-type discretization theorem with parameters m and q if there exist a set of knots {ξν ∈ Ω}, a set of weights {λν}, ν = 1, . . . , m, and two positive constants Cj(d, q), j = 1, 2, such that for any f ∈ XN we have C1(d, q)f q

q ≤ m

• ν=1

λν|f (ξν)|q ≤ C2(d, q)f q

q.

(3) Then we also say that the Mw(m, q) theorem holds for a subspace XN or XN ∈ Mw(m, q). Obviously, XN ∈ M(m, q) implies that XN ∈ Mw(m, q).

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Marcinkiewicz problem with ε

We write XN ∈ M(m, q, ε) if (1) holds with C1(d, q) = 1 − ε and C2(d, q) = 1 + ε. Respectively, we write XN ∈ Mw(m, q, ε) if (3) holds with C1(d, q) = 1 − ε and C2(d, q) = 1 + ε.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Marcinkiewicz problem with ε

We write XN ∈ M(m, q, ε) if (1) holds with C1(d, q) = 1 − ε and C2(d, q) = 1 + ε. Respectively, we write XN ∈ Mw(m, q, ε) if (3) holds with C1(d, q) = 1 − ε and C2(d, q) = 1 + ε. We note that the most powerful results are for M(m, q, 0), when the Lq norm of f ∈ XN is discretized exactly by the formula with equal weights 1/m.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

More general setting

Sampling discretization is a natural way of estimating the quantity

• f interest f q
• q. Certainly, one can ask a question of optimal

estimation of f q

q using m function values or, even more general,

using m linear functionals. It is an interesting problem but we do not address it in this talk. We only point out on a simple example that we obtain very different results when we allow arbitrary linear functionals to be used.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Example

Consider the class Wr

2 of periodic functions with bounded in L2 the

rth mixed derivative. For N ∈ N define the hyperbolic cross Γ(N) := {k = (k1, . . . , kd) ∈ Zd :

d

• j=1

max(1, |kj|) ≤ N} and for f ∈ L1 SN(f ) :=

• k∈Γ(N)

ˆ f (k)ei(k,x), ˆ f (k) := (2π)−d

• [0,2π]d f (x)e−i(k,x)dx.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Example continue

Then, it is well known and easy to prove that for any f ∈ Wr

2 we

have 0 ≤ f 2

2 − SN(f )2 2 ≤ N−2r.

(4) With this algorithm we use m ≍ N(log N)d−1 linear functionals ˆ f (k), k ∈ Γ(N). The bound (4) is very different from the asymptotic behavior in Theorem T2.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

A remark on general inequalities

The sampling discretization errors erm(W , Lq) and ero

m(W , Lq)

are new asymptotic characteristics of a function class W .

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

A remark on general inequalities

The sampling discretization errors erm(W , Lq) and ero

m(W , Lq)

are new asymptotic characteristics of a function class W . It is natural to try to compare these characteristics with other classical asymptotic characteristics.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

A remark on general inequalities

The sampling discretization errors erm(W , Lq) and ero

m(W , Lq)

are new asymptotic characteristics of a function class W . It is natural to try to compare these characteristics with other classical asymptotic characteristics. Theorem T1 addresses this issue. It is known that the sequence of entropy numbers is one of the smallest sequences

• f asymptotic characteristics of a class. For instance, by Carl’s

inequality it is dominated, in a certain sense, by the sequence

• f the Kolmogorov widths.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

A remark on general inequalities continue

Theorem T1 shows that the sequence {εn(W )} dominates, in a certain sense, the sequence {erm(W )}.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

A remark on general inequalities continue

Theorem T1 shows that the sequence {εn(W )} dominates, in a certain sense, the sequence {erm(W )}. Clearly, alike the Carl’s inequality, one tries to prove the corresponding relations in as general situation as possible.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

A remark on general inequalities continue

Theorem T1 shows that the sequence {εn(W )} dominates, in a certain sense, the sequence {erm(W )}. Clearly, alike the Carl’s inequality, one tries to prove the corresponding relations in as general situation as possible. We derive Theorem T1 from known results in learning theory. Our proof is a probabilistic one. The use of that kind of technique results in the limitation r ∈ (0, 1/2) for the power in the rate of decay of the entropy numbers. As we pointed

• ut above, we do not know if one can prove an analog of

Theorem T1 in the case r > 1/2.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

T1 is optimal. Upper bound

Known results on the asymptotic characteristics of the univariate class Wr

p show that Theorem T1 cannot be improved. Namely, on

• ne hand it is known that

εn(Wr

p, L∞) ≍ n−r,

r > 1/p, 1 ≤ p ≤ ∞. (5)

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

T1 is optimal. Upper bound

Known results on the asymptotic characteristics of the univariate class Wr

p show that Theorem T1 cannot be improved. Namely, on

• ne hand it is known that

εn(Wr

p, L∞) ≍ n−r,

r > 1/p, 1 ≤ p ≤ ∞. (5) For 2 < p < ∞ and r ∈ (1/p, 1/2) relation (5) and Theorem T1 imply erm(Wr

p) ≤ C(r, p)m−r.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

T1 is optimal. Lower bound

On the other hand, assume that a class of real functions W ⊂ C(Ω) has the following extra property. Property A. For any f ∈ W we have f + := (f + 1)/2 ∈ W and f − := (f − 1)/2 ∈ W .

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

T1 is optimal. Lower bound

On the other hand, assume that a class of real functions W ⊂ C(Ω) has the following extra property. Property A. For any f ∈ W we have f + := (f + 1)/2 ∈ W and f − := (f − 1)/2 ∈ W . In particular, this property is satisfied if W is a convex set containing function 1.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

T1 is optimal. Lower bound

On the other hand, assume that a class of real functions W ⊂ C(Ω) has the following extra property. Property A. For any f ∈ W we have f + := (f + 1)/2 ∈ W and f − := (f − 1)/2 ∈ W . In particular, this property is satisfied if W is a convex set containing function 1. For a function class W ⊂ C(Ω) consider the best error of numerical integration by cubature formulas with m knots: κm(W ) := inf

(ξ,Λ) sup f ∈W

|Iµ(f ) − Λm(f , ξ)|, Iµ(f ) :=

• Ω

fdµ, Λm(f , ξ) :=

m

• j=1

λjf (ξj).

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Connection to numerical integration

Theorem (T3; VT, 2018) Suppose W ⊂ C(Ω) has Property A. Then for any m ∈ N we have ero

m(W , L2) ≥ 1

2κm(W ).

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Lower bound

It is known that κn(Wr

p) ≍ n−r,

r > 1/p, 1 ≤ p ≤ ∞. (6) Theorem T3 and relation (6) imply erm(Wr

p) ≥ C(r, p)m−r.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Connection to learning theory

Let X ⊂ Rd, Y ⊂ R be Borel sets, ρ be a Borel probability measure on a Borel set Z ⊂ X × Y . For f : X → Y define the error E(f ) :=

• Z

(f (x) − y)2dρ.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Connection to learning theory

Let X ⊂ Rd, Y ⊂ R be Borel sets, ρ be a Borel probability measure on a Borel set Z ⊂ X × Y . For f : X → Y define the error E(f ) :=

• Z

(f (x) − y)2dρ. Let ρX be the marginal probability measure of ρ on X, i.e., ρX(S) = ρ(S × Y ) for Borel sets S ⊂ X. Define fρ(x) := E(y|x) to be a conditional expectation of y.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Setting

The function fρ is known in statistics as the regression function of ρ. In the sense of error E(·) the regression function fρ is the best to describe the relation between inputs x ∈ X and outputs y ∈ Y .

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Setting

The function fρ is known in statistics as the regression function of ρ. In the sense of error E(·) the regression function fρ is the best to describe the relation between inputs x ∈ X and outputs y ∈ Y . The goal is to find an estimator fz, on the base of given data z := ((x1, y1), . . . , (xm, ym)) that approximates fρ well with high probability.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Setting

The function fρ is known in statistics as the regression function of ρ. In the sense of error E(·) the regression function fρ is the best to describe the relation between inputs x ∈ X and outputs y ∈ Y . The goal is to find an estimator fz, on the base of given data z := ((x1, y1), . . . , (xm, ym)) that approximates fρ well with high probability. We assume that (xi, yi), i = 1, . . . , m are independent and distributed according to ρ.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Setting

The function fρ is known in statistics as the regression function of ρ. In the sense of error E(·) the regression function fρ is the best to describe the relation between inputs x ∈ X and outputs y ∈ Y . The goal is to find an estimator fz, on the base of given data z := ((x1, y1), . . . , (xm, ym)) that approximates fρ well with high probability. We assume that (xi, yi), i = 1, . . . , m are independent and distributed according to ρ. We measure the error between fz and fρ in the L2(ρX) norm.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Entropy numbers

For a compact subset Θ of a Banach space B we define the entropy numbers as follows εn(Θ, B) := inf{ε : ∃f1, . . . , f2n ∈ Θ : Θ ⊂ ∪2n

j=1(fj + εU(B))}

where U(B) is the unit ball of a Banach space B.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Empirical error

We define the empirical error of f as Ez(f ) := 1 m

m

• i=1

(f (xi) − yi)2.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Empirical error

We define the empirical error of f as Ez(f ) := 1 m

m

• i=1

(f (xi) − yi)2. Let f ∈ L2(ρX). The defect function of f is Lz(f ) := Lz,ρ(f ) := E(f ) − Ez(f ); z = (z1, . . . , zm), zi = (xi, yi).

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Empirical error

We define the empirical error of f as Ez(f ) := 1 m

m

• i=1

(f (xi) − yi)2. Let f ∈ L2(ρX). The defect function of f is Lz(f ) := Lz,ρ(f ) := E(f ) − Ez(f ); z = (z1, . . . , zm), zi = (xi, yi). We are interested in estimating Lz(f ) for functions f coming from a given class W . We assume that ρ and W satisfy the following condition: for all f ∈ W and any (x, y) ∈ Z |f (x) − y| ≤ M. (7)

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Estimate for the defect function

Theorem (S. Konyagin and VT, 2004) Assume ρ, W satisfy (7) and εn(W , L∞) ≤ Dn−r, r ∈ (0, 1/2). Then for m, η satisfying mη1/r ≥ C1(M, D, r) we have ρm{z : sup

f ∈W

|Lz(f )| ≥ η} ≤ C(M, D, r) exp(−c(M, D, r)mη1/r).

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

An interesting phenomenon

There are results (see G.W. Wasilkowski, 1984) on optimal estimation of the f under assumption that f ∈ W . At a first glance the problems of estimation of f and, say, estimation of f 2, like in our case, are very close.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

An interesting phenomenon

There are results (see G.W. Wasilkowski, 1984) on optimal estimation of the f under assumption that f ∈ W . At a first glance the problems of estimation of f and, say, estimation of f 2, like in our case, are very close. A simple inequality |a2 − b2| ≤ 2M|a − b| for numbers satisfying |a| ≤ M and |b| ≤ M shows that normally we can get an upper bound for estimation of f 2 in terms of the error of estimation of f . However, it turns out that the above two problems are different.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

An interesting phenomenon continue

It is proved in G.W. Wasilkowski, 1984 that the error of

• ptimal estimation of the · is of the same order as the
• ptimal error of approximation. For instance, in case of the

class Wr

2 this error is of the order m−r(log m)r(d−1), which is

larger than the corresponding error ero

m(Wr 2, L2) in Theorem

T2.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

An interesting phenomenon continue

It is proved in G.W. Wasilkowski, 1984 that the error of

• ptimal estimation of the · is of the same order as the
• ptimal error of approximation. For instance, in case of the

class Wr

2 this error is of the order m−r(log m)r(d−1), which is

larger than the corresponding error ero

m(Wr 2, L2) in Theorem

T2. The above Example shows that the optimal error for estimation of the f may be different from the optimal error

• f estimation of the f 2.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

An interesting phenomenon continue

It is proved in G.W. Wasilkowski, 1984 that the error of

• ptimal estimation of the · is of the same order as the
• ptimal error of approximation. For instance, in case of the

class Wr

2 this error is of the order m−r(log m)r(d−1), which is

larger than the corresponding error ero

m(Wr 2, L2) in Theorem

T2. The above Example shows that the optimal error for estimation of the f may be different from the optimal error

• f estimation of the f 2.

Detailed comparison of my paper with G.W. Wasilkowski, 1984 shows that the problems of optimal errors in estimation

• f f and f 2 are different.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Quasi-algebra property.

We begin with a very simple general observation on a connection between norm discretization and numerical integration. Quasi-algebra property. We say that a function class W has the quasi-algebra property if there exists a constant a such that for any f , g ∈ W we have fg/a ∈ W .

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Quasi-algebra property.

We begin with a very simple general observation on a connection between norm discretization and numerical integration. Quasi-algebra property. We say that a function class W has the quasi-algebra property if there exists a constant a such that for any f , g ∈ W we have fg/a ∈ W . The above property was introduced and studied in detail by H.

• Triebel. He introduced this property under the name multiplication
• algebra. Normally, the term algebra refers to the corresponding

property with parameter a = 1. To avoid any possible confusions we call it quasi-algebra. We refer the reader to the very resent book of Triebel, 2018, which contains results on the multiplication algebra (quasi-algebra) property for a broad range of function spaces.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Upper bound

Proposition (P1; VT, 2018) Suppose that a function class W has the quasi-algebra property and for any f ∈ W we have for the complex conjugate function ¯ f ∈ W . Then for a cubature formula Λm(·, ξ) we have: for any f ∈ W |f 2

2 − Λm(|f |2, ξ)| ≤ a sup g∈W

• Ω

gdµ − Λm(g, ξ)

• .

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Function classes

We discuss some classical classes of smooth periodic functions. We begin with a general scheme and then give a concrete example. Let F ∈ L1(Td) be such that ˆ F(k) = 0 for all k ∈ Zd, where ˆ F(k) := F(F, k) := (2π)−d

• Td F(x)e−i(k,x)dx.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Function classes

We discuss some classical classes of smooth periodic functions. We begin with a general scheme and then give a concrete example. Let F ∈ L1(Td) be such that ˆ F(k) = 0 for all k ∈ Zd, where ˆ F(k) := F(F, k) := (2π)−d

• Td F(x)e−i(k,x)dx.

Consider the space W F

2 := {f : f (x) = JF(ϕ)(x) := (2π)−d

• Td F(x − y)ϕ(y)dy,

ϕ2 < ∞}.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Quasi-algebra property for function classes

For f ∈ W F

2 we have ˆ

f (k) = ˆ F(k) ˆ ϕ(k) and, therefore, our assumption ˆ F(k) = 0 for all k ∈ Zd implies that function ϕ is uniquely defined by f . Introduce a norm on W F

2 by

f W F

2 := ϕ2,

f = JF(ϕ).

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Quasi-algebra property for function classes

For f ∈ W F

2 we have ˆ

f (k) = ˆ F(k) ˆ ϕ(k) and, therefore, our assumption ˆ F(k) = 0 for all k ∈ Zd implies that function ϕ is uniquely defined by f . Introduce a norm on W F

2 by

f W F

2 := ϕ2,

f = JF(ϕ). For convenience, with a little abuse of notation we use notation W F

2 for the unit ball of the space W F 2 . We are interested in the

following question. Under what conditions on F the fact that f , g ∈ W F

2 implies that fg ∈ W F 2 and

fgW F

2 ≤ C0f W F 2 gW F 2 ? Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Quasi-algebra property for function classes

For f ∈ W F

2 we have ˆ

f (k) = ˆ F(k) ˆ ϕ(k) and, therefore, our assumption ˆ F(k) = 0 for all k ∈ Zd implies that function ϕ is uniquely defined by f . Introduce a norm on W F

2 by

f W F

2 := ϕ2,

f = JF(ϕ). For convenience, with a little abuse of notation we use notation W F

2 for the unit ball of the space W F 2 . We are interested in the

following question. Under what conditions on F the fact that f , g ∈ W F

2 implies that fg ∈ W F 2 and

fgW F

2 ≤ C0f W F 2 gW F 2 ?

In other words: Which properties of F guarantee that the class W F

2 has the quasi-algebra property? We give a simple sufficient

condition.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Sufficient condition for quasi-algebra prperty

Proposition (P2; VT, 2018) Suppose that for each n ∈ Zd we have

• k∈Zd

| ˆ F(k) ˆ F(n − k)|2 ≤ C 2

0 | ˆ

F(n)|2. (8) Then, for any f , g ∈ W F

2 we have fg ∈ W F 2 and

fgW F

2 ≤ C0f W F 2 gW F 2 . Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Classes with mixed smoothness

As an example consider the class Wr

2 of functions with bounded

mixed derivative. By the definition Wr

2 := W Fr 2

with function Fr(x) defined as follows. For a number k ∈ Z denote k∗ := max(|k|, 1). Then for r > 0 we define Fr by its Fourier coefficients ˆ Fr(k) =

d

• j=1

(k∗

j )−r.

(9)

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Classes with mixed smoothness

As an example consider the class Wr

2 of functions with bounded

mixed derivative. By the definition Wr

2 := W Fr 2

with function Fr(x) defined as follows. For a number k ∈ Z denote k∗ := max(|k|, 1). Then for r > 0 we define Fr by its Fourier coefficients ˆ Fr(k) =

d

• j=1

(k∗

j )−r.

(9) Lemma (L1) Function F = Fr with r > 1/2 satisfies condition (8) with C0 = C(r, d).

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Classes with mixed smoothness

As an example consider the class Wr

2 of functions with bounded

mixed derivative. By the definition Wr

2 := W Fr 2

with function Fr(x) defined as follows. For a number k ∈ Z denote k∗ := max(|k|, 1). Then for r > 0 we define Fr by its Fourier coefficients ˆ Fr(k) =

d

• j=1

(k∗

j )−r.

(9) Lemma (L1) Function F = Fr with r > 1/2 satisfies condition (8) with C0 = C(r, d). Lemma L1 and Proposition P1 imply that the class Wr

2 has the

quasi-algebra property.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Fibonacci cubature formulas

We now illustrate how a combination of Proposition P1 and known results on numerical integration gives results on discretization. We discuss classes of periodic functions of two variables. Let {bn}∞

n=0,

b0 = b1 = 1, bn = bn−1 + bn−2, n ≥ 2, – be the Fibonacci numbers.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Fibonacci cubature formulas

We now illustrate how a combination of Proposition P1 and known results on numerical integration gives results on discretization. We discuss classes of periodic functions of two variables. Let {bn}∞

n=0,

b0 = b1 = 1, bn = bn−1 + bn−2, n ≥ 2, – be the Fibonacci

• numbers. For continuous functions of two variables, which are

2π-periodic in each variable, we define cubature formulas Φn(f ) := b−1

n bn

• µ=1

f

• 2πµ/bn, 2π{µbn−1/bn}
• ,

which are called the Fibonacci cubature formulas. In this definition {a} is the fractional part of the number a.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Known result for numerical integration

For a function class W denote Φn(W ) := sup

f ∈W

|Φn(f ) − ˆ f (0)|. The following result is known Φn(Wr

2) ≍ b−r n (log bn)1/2,

r > 1/2. (10) Combining (10) with Proposition P1 we obtain the following discretization result.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

Discretization result

Theorem (T4; VT, 2018) Let d = 2, r > 1/2 and µ be the Lebesgue measure on [0, 2π]2. Then erm(Wr

2, L2) ≤ C(r)m−r(log m)1/2.

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3

The End

Thank you!

Vladimir Temlyakov Sampling discretization of integral norms. Lecture 3